COMP ENG 4TL4: Digital Signal Processing Notes for Lecture #3 Wednesday, September 10, 2003
1.4 Quantization Digital systems can only represent sample amplitudes with a finite set of prescribed values, and thus it is necessary for A/D converters to quantize the values of the samples x[n]. A typical form of quantization uses uniform quantization steps, where the input voltage is either rounded or truncated. (Porat)
A 3-bit rounding uniform quantizer with quantization steps of is shown below. (Oppenheim and Schafer)
Two forms of quantization error e[n] exist: 1. Quantization noise: due to rounding or truncation over the range of quantizer outputs; and 2. Saturation ( peak clipping ): due to the input exceeding the maximum or minimum quantizer output. Both types of error are illustrated for the case of a sinusoidal signal in the figure on the next slide. Quantization noise can be minimized by choosing a sufficiently small quantization step. We will derive a method for quantifying how small is sufficient. Saturation can be avoided by carefully matching the fullscale range of an A/D converter to anticipated input signal amplitude ranges.
Saturation or peak clipping (Oppenheim and Schafer)
For rounding uniform quantizers, the amplitude of the quantization noise is in the range /2 < e[n] /2. For small it is reasonable to assume that e[n] is a random variable uniformly distributed over ( /2, /2]. For fairly complicated signals, it is reasonable to assume that successive quantization noise values are uncorrelated and that e[n] is uncorrelated with x[n]. Thus, e[n] is assumed to be a uniformly distributed whitenoise sequence with a mean of zero and variance: For a (B+1)-bit quantizer with full-scale X m, the noise variance (or power) is:
The signal-to-noise ratio (SNR) of a (B+1)-bit quantizer is: where σ x2 is the signal variance (or power). Analog signals such as speech and music typically have a Gaussian (or super-gaussian) amplitude distribution, and consequently samples rarely exceed 3 or 4 times the standard deviation. To avoid saturation (peak clipping) we might set X m = 4σ x, in which case:
This uniform quantization scheme is called pulse code modulation (PCM). Advantages: no coding delay not signal specific Disadvantages: high bit rates e.g. Wireless telephony requires 11 bits for toll quality analog telephone quality. If f s = 10,000 Hz bit rate = 110,000 bps, which may be impractical for wireless systems. However, consider a CD player that uses 16-bit PCM SNR 88.75 db & bit rate 320,000 bps, which is acceptable for wired applications.
Uniform quantization is suboptimal for many applications. Consider the probability density function (p.d.f.) of speech: (Quatieri)
If the p.d.f. of signals to be quantized is known, then an optimal nonuniform quantization scheme can be derived: (Quatieri)
Sometime the p.d.f. of signals to be quantized is known (or can be assumed), but the signal variance (power) may vary over time. In this case, an adaptive nonuniform quantization scheme can be employed: (Quatieri)
An alternative to nonuniform quantization is companding, in which a nonlinearity is used to produce a new discrete-time signal that has a uniform distribution, and a uniform quantizer can consequently be applied: (Quatieri)
Other nonlinearities approximate the companding operation, but are easier to implement and do not require a p.d.f. measurement. One example is µ-law companding, ubiquitous in waveform coding, which uses the nonlinearity: Together with uniform quantization, this nonlinearity (for large values of µ, e.g., 255) yields an SNR approximately independent of X m and σ x over a large range of signal input. For example, toll quality speech is obtainable with µ-law PCM utilizing µ-law companding followed by a 7-bit uniform quantizer, which would require 11-bit quantization without the companding operation.
(Quatieri)